# Page:Lawhead columbia 0054D 12326.pdf/139

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Moreover, we can deduce a second balance equation for the atmosphere alone. Recall that the atmosphere receives energy only from the surface, and that it radiates with twice the area that it receives—it is "heated" from below only, but radiates heat in two directions. With another application of the Stefan-Boltzmann law, then, we know that:

 ${\displaystyle 2\gamma _{a}\sigma T_{a}^{4}=\gamma _{s}\sigma T_{s}^{4}}$ (4j)

A bit of algebraic manipulation to solve this system of equation—by inserting (4j) into (4f) and solving the resulting equation for Ts—gives us a final solution to the whole shebang (as noted above, we shall assume that the Earth is opaque and that γs = 1):

 ${\displaystyle {\sqrt[{4}]{\tfrac {S_{o}(1-\sigma )}{4\sigma (1-{\frac {\gamma _{a}}{2}})}}}=T_{s}}$ (4k)

With no atmosphere at all, ${\displaystyle \gamma _{a}=0}$ and the equation above just reduces to our original equation, giving us an answer of 255K. By plugging in the observed temperature at the Earth's surface (288K) and solving for ${\displaystyle \gamma _{a}}$, we obtain a value of ${\displaystyle \gamma =.76}$. With that value in hand, then, we can actually use this model to explore the response of the planet to changes in albedo or greenhouse gas composition—we can make genuine predictions about what will happen to the planet if our atmosphere becomes more opaque to infrared radiation, more energy comes in from the sun, or the reflective profile of the surface changes. This is a fully-developed ZDEBM, and while it is only modestly powerful, it is a working model that could be employed to make accurate, interesting predictions. It is a real pattern.

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